Which of the following is the graph of <span class="math-inline">\(y=\frac{\sin x}{\sin x}\)</span> ?
Step-by-Step Solution
Key Concept: The function y = sin(x)/sin(x) is not simply equal to 1 for all x; we must carefully consider the domain where sin(x) ≠ 0. The function is undefined wherever sin(x) = 0, creating discontinuities at x = nπ for all integers n.
<p><strong>Step 1:</strong> Analyze the function y = sin(x)/sin(x).</p><p><strong>Step 2:</strong> Identify the domain: sin(x) ≠ 0, which means x ≠ nπ for any integer n. The points x = 0, ±π, ±2π, ±3π, ... are excluded from the domain.</p><p><strong>Step 3:</strong> For all x in the domain (where sin(x) ≠ 0), we can simplify: y = sin(x)/sin(x) = 1.</p><p><strong>Step 4:</strong> The graph is a horizontal line at y = 1, but with holes (open circles) at x = nπ for every integer n, since the function is undefined at these points.</p><p><strong>Step 5:</strong> Graph B correctly shows a horizontal line at y = 1 with open circles/holes at x = 0, ±π, ±2π, etc., indicating the discontinuities in the domain.</p><p><strong>∴ Answer:</strong> B</p>
Correct Answer: B